A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.     

An algorithmic approach to solving polynomial equations associated with quantum circuits

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F ₂ to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F ₂ whose variables also take values in F ₂ (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F ₂ associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F ₂.     

On the finiteness of scaling sum rules

The convergence of sum rules relating the matrix elements of local operators to integrals over deep-inelastic structure functions is studied critically. It is found that the matrix elements may always be written as the q² → ? ∞ limit of finite expressions, regardless of the (Regge) asymptotic behavior of the structure functions or the possible occurrence of J = 0 fixed singularities. The correct form of the sum rule for the operator Schwinger term is taken as a paradigm case. It is derived from the Bjorken-Johnson-Low theorem and agrees with the results of parton model and light-cone analyses. It readily encompases the results of second order φ³ theory (where the Schwinger term diverges logarithmically) and second order vector gluon theory (where it vanishes). Sufficient conditions for the finiteness of the operator Schwinger term are the scaling of the longitudinal structure function and the absence of J = 0 fixed singularities with nonpolynomial residues. The treatment is readily applicable to other scaling and fixed q² sum rules needing regulation. A compendium of these is given.     

The geometry of minimal replacement for the Poincaré group

The constructs of this paper rest on two elementary facts: (1) the Poincaré group P₁₀ is the maximal group of isometries of Minkowski space-time M₄; (2) P₁₀ has a faithful matrix representation as a subgroup of GL(5, R) that maps an affine set into itself. Local action of P₁₀ and Yang-Mills minimal replacement are shown to induce a well-defined minimal replacement operator that maps the tensor algebra over M₄ onto the tensor algebra over a new space-time U₄. The natural frame and coframe fields of M₄ go over into a canonical system of frame and coframe fields of U₄ with both translation and Lorentz-rotation parts. The coframe fields define soldering 1-form fields for U₄ that give rise to the standard geometric quantities through the Cartan equations of structure. This leads to unique determinations of all relevant connection coefficients and the associated 2-forms of curvature and torsion that involve the compensating 1-forms for local action of both the translation and the Lorentz-rotation sectors. The metric tensor of U₄, that is induced by the minimal replacement operator, is shown to satisfy the Ricci lemma; U₄ is necessarily a Riemann-Cartan space. This space admits gauge covariant constant basis fields for the Lie algebra of the Lorentz group and for the Dirac algebra. The induced basis for the Dirac algebra evaluates the images of Dirac operators under minimal replacement, while the induced basis for the Lie algebra of L(4, R) serves to show that the holonomy group of U₄ is the Lorentz group. The minimal replacement operator is extended to include the case of a total gauge group that is the direct product of the Poincaré group and a Lie group of internal symmetries of matter fields. This provides a precise method of lifting any action integral of the matter fields from M₄ up to U₄ so that invariance properties are retained when the total group acts locally. The natural representations afforded by minimal replacement result in curvature being evaluated in terms of first order derivatives of the compensating fields that share many properties in common with the Dirac derivation algebra for spin fields. Direct interpretations of the compensating fields are obtained from the geodesic equations.     

Quantum algebra U q(2, 1): q analogs of the Gelfand-Graev formulas

The discrete series of unitary irreducible representations of the noncompact quantum algebra U _q(2, 1) are studied. For the negative discrete series, two bases of these irreps are considered. One of them corresponds to the reduction U _q(2, 1) → U _q(2)×U(1). The second basis is connected with the reduction U _q(2, 1) → U(1)×U _q(1, 1). The matrix elements of the U _q(2, 1) generators in both bases are calculated. For the intermediate discrete series, only first type of basis is considered and the q analogs of the Gelfand-Graev formulas are obtained. Also, the transformation brackets connecting the two bases are found for the negative discrete series.     

Manifestation of higher polarization moments in the dipole emission under conditions of stepwise excitation and anisotropic collisions

The induction of higher polarization moments-the octupole orientation and the hexadecapole alignment—in the case of two-step excitation of atoms through dipole transitions and their manifestation in the polarization of dipole emission under conditions of anisotropic collisions are studied. For all channels of the two-step excitation process J ₀→J ₁ J, the efficiencies of production of higher polarization moments with integer and half-integer angular momenta from J=2 to J=7 through the absorption of linearly or circularly polarized light are calculated. The rates of collisional relaxation and interconversion of higher polarization moments, for both orientation and alignment, as functions of frequency detuning of the laser line from the center of the Doppler profile are calculated. The theory is illustrated by an example of two channels of excitation of the atomic state J=2 by circularly polarized light: J ₀=0→J ₁=1J=2 and J ₀=3→J 1=2 (or 3)→J=2. In the first case, the octupole orientation enhances or attenuates the signal of circular polarization depending on the laser frequency detuning. In the second case, it represents the only source producing orientation and circular polarization of the emitted light.     

u q(2,1) noncompact quantum algebra: Intermediate discrete series of unitary irreducible representations

The unitary irreducible representations of the u _q(2,1) quantum algebra that belong to the intermediate discrete series are considered. The q analog of the Mickelsson-Zhelobenko algebra is developed. Use is made of the U basis corresponding to the reduction u _q(2,1) ? u _q(2). Explicit formulas for the matrix elements of the generators are obtained in this basis. The projection operator that projects an arbitrary vector onto the extremal vector of the intermediate-series representation is found.     

On the associativity anomaly in open string field theory

The associativity anomaly in the star algebra of the open bosonic string is demonstrated by a simple oscillator calculation. In an associative algebra, formal arguments require that the “half string” BRST operator Q_L be nilpotent; however, we show that associativity is actually violated by computing Q_L² in an explicit operator representation of the star algebra.     

Lie-theoretic generating relations of two variable Laguerre polynomials

Generating relations involving two variable laguerre polynomials L_n(x, y) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L_n(x, y).     

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W_n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.     

Hard pions and axial meson exchange currents in nuclear physics

Starting from the hard meson method we develop a consistent approach to the problem of the axial meson exchange currents (MEC). This method incorporates the current algebra and PCAC together with the vector dominance and allows one to study the pion as well as heavy-meson exchanges on an equal footing. Using a minimal, chiral and approximately gauge-invariant phenomenological Lagrangian (PL) model for the A_1ρπ system we construct the two-nucleon axial MEC operator in the tree approximation. This operator automatically possesses the correct chiral SU(2) × SU(2) transformation properties and has the smoothest momentum dependence which is allowed within the combined current algebra and vector dominance approaches. In the given model, we consider the non-Born part of the amplitude N + J^A → n + π and demonstrate that in the soft pion limit, it exactly coincides with the PCAC prediction.     

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra su_q(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.     

A New Current Algebra and the Reflection Equation

We establish an explicit algebra isomorphism between the quantum reflection algebra for the U_q([^(sl₂)]) R{U_q(\widehat{sl_2}) R}-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).     

NMR and magnetic susceptibility of CeCu5−xNix intermetallic compounds

The ⁶³Cu Knight shift and magnetic susceptibility of the CeCu_5?xNi_x intermetallic system have been investigated as a function of temperature. The compounds studied were all paramagnets in the temperature range 90–700K with θ_p < 0. The results obtained reveal a reversal s-f coupling with positive J_sf and J_sd coupling constants.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

On the breaking of chiral symmetry in the Schwinger model

The Schwinger model is reinvestigated in the A₁ = 0 gauge. Based on the explicit operator solution, it is suggested that the breaking of chiral symmetry is dynamical and the vacuum is unique contrary to the arguments by Kogut and Susskind.     

Some Functions that Generalize the Askey–Wilson Polynomials

We determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q-difference operator. The solution is described in terms of an arbitrary solution of a q-analogue of Gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the Askey-Wilson polynomials. There is some evidence that this bispectral problem, for an arbitrary order q-difference operator, is intimately related with some q-deformation of the Toda lattice hierarchy and its Virasoro symmetries. When tridiagonal matrices are replaced by the Schroedinger operator, and q= 1, this statement holds with Toda replaced by KdV. In this context, this paper determines the analogs of the Bessel and Airy potentials. Received: 7 May 1996/Accepted: 30 August 1996     

Impurity 55Mn NMR in Cu-Mn dilute alloys

We detected impurity ⁵⁵Mn NMR in Cu-45 ppm and -110 ppm Mn dilute alloys at temperatures of 150–330 K. The hyperfine field we obtained was 65±5 kOe/μ_B. From a temperature variation of linewidths we could estimate J_sd as 1.2±0.1 eV, using the reported theories of relaxation times T₁ and T₂.